FOM: use and mention, objective, astronomical calculations, etc.

[Vaughan Pratt]

From: "Moshe' Machover" <moshe.machover at kcl.ac.uk>
>Humans make mathematics, but they do not make it as they please. Once
>mathematical objects are invented, they obey ineluctable laws. It is this
>*fact* that creates the illusion of platonism, and enables mathematicians
>to think and reason *as though* matheamtical objects exist independently
>of anyone's consciousness.

Yes!  This is the essence of my point in March #329, where I said

>I would compare mathematics to diamond jewelry.  You can't substitute
>glass, the experts know the difference and you have to hunt for the real
>stuff and use only what nature gives you.  But neither can you say that
>the jewelry was there in the earth all along.

Humans make jewelry, but they do not make it as they please (if they
want it to be taken seriously).

Interestingly Hersh's reaction to my diamond jewelry analogy, March #333,
was a one-word posting: "yes!"  Yet nowhere does Hersh acknowledge the
mathematical counterpart of "you can't substitute glass".  Reuben, could
you comment on this, as well as on Machover's "Humans make mathematics,
but they do not make it as they please"?

From: Thomas Forster <T.Forster at dpmms.cam.ac.uk>
>Vaughan, I don't think you have to go to other planets to get instances of the 
>kind you want.  Just the other side of a tunnel from the island i'm on
>lies a land inhabited by a stange tribe whose philosophy, views on
>lierary theory etc, etc, are radically different from those on this island
>but whose mathematics is very similar...

Three points.  First, a literary critic might be forgiven for reading
FOM and coming to the conclusion that views on mathematics are no less
disparate than views on literary criticism.  Indeed if she follows the
reasoning in the literary arguments but not in the mathematical ones
she may form the opposite impression to yours.  (Rats, where are the
literary critics when you need one?)

Second, we have Steve Simpson's post of October #24,

>I know a little about topos theory, but I have never really understood
>the way category theorists seem to view the rest of mathematics, not
>to mention the rest of human knowledge,

witnessing that at least some people already find radically different
mathematics here on earth, sufficiently different in fact as to make
them deny that it counts as basic mathematics.  (I used this posting in
November #3 to make the point that Martin Gardner didn't need to go to
another planet to "learn that little green critters, with textbooks full
of mysterious definitions and theorems purporting to be the true view of
mathematics, are already among us and even holding conferences in nice
places like Montreal and Vancouver where they engage in extraterrestrial
mathematics.")

Third, why should the situation on other planets resemble that on earth?
In 1940 no one was predicting category theory even as a mathematical
subject let alone as an upstart challenger to set theory.  Why should
we believe we have now discovered the whole gamut of radically different
mathematics?  For all we know category theory may be much closer to set
theory than to Arcturan mathematics.

Arcturus is however nearby, and on its planets we can at least reasonably
assume that the *physics* is the same.  But what evidence do we have,
beyond beliefs and wishful calculations, that physics remains the same on
the antipodal side of the universe?  Wherever the physics is radically
different, the mathematics may be even more different there than on our
putative inhabited Arcturan planet.

Vaughan Pratt